It is pointed out that the (bosonic) symmetries of the superpotential term is an important notion in supersymmetric theories. Nambu-Goldstone like theorems are proved for the spontaneous breaking of the superpotential symmetries, under the assumption that supersymmetry itself remains unbroken.

It is well known that quite often in supersymmetric theories many unexpected pseudo Nambu-Goldstone particles appear when internal symmetries break down spontaneously [ 1 ]. The origin of such phenomena, however, has never been understood clearly.

We show in this letter that the notion of symmetries ofsuperpotential term has much importance in this context: The spontaneous breaking of superpotential symmetries explains the appearance of such unexpect- ed massless particles in almost all the cases. Indeed the superpotential term has necessarily a much larger sym- metry than the full lagrangian since the former is a function of chiral multiplets alone while the latter contains anti-chiral multiplets Cas well as . So, for instance, when the lagrangian is invariant under a real Lie group GO, the superpotential term is necessarily in- variant under the complex form G~ of G O at least. This has been known probably to many authors [2-5] . The purpose of this note is to prove that Nambu- Goldstone like theorems hold for such superpotential symmetries. Actually, Lee and Sharatchandra [4] and Lerche [5] have mentioned such a type of theorem for the first time correctly referring to the super-potential

symmetries. Unfortunately, since they did not recog- nize sufficiently the importance of the notion of super- potential symmetry, however, their "theorem" was not so clearly written and further was restricted to the simple systems consisting of chiral supermultiplets alone. Therefore we here generalize their result to ar- bitrary systems with global (N = 1) supersymmetry and present the theorems in a clear and complete form.

Before going into the most general cases, we first concentrate on simpler systems consisting of chiral multiplets ~i(x, O) alone. The general form of the lagrangian for such a system is given by ,1

= f d2Od2Of (~,)+( fd2Og()+h.c . ) . (1)

We assume that the superpotential g() is invariant under transformations of a symmetry group G(C):

(e)~i ~ [~)i, 6A XA ] = eA A/A (), (2)

8 (e)g() = e A (~g/~i) ~A (~) = 0 . (3)

Here X A's are the generators of G(C) and AA() take the simple forms

At" A (4) = ( TA)/], TA: repr. matrix of X A , (4)

+1 For notations and conventions we follow the book of Wess and Bagger [3.]

when X A -transformations are linear but may be non- linear functions of q) in general. Notice that the sym- metry group G(C) of superpotential is necessarily a o)mplex group as mentioned above; i.e., the transfor- mation parameters e A are complex, e A @ C. Indeed, if the invariance eq. (3) holds for real eA, it does also for complex CA, since (Og/~q)i) AA(q)) = 0 follow from the fl)rmer. We use the notation G(C) to emphasize this point.

We confine ourselves to the cases where the super- symmetry is not broken spontaneously, which in particular implies that the vacuum expectation value (VEV) of the superfields q)i

%(x, 0))= v;, (5)

is 0-independent (and also x-independent by the trans- lational invariance). We assume that the system (1) is non-singular and has positive metric, at least on the vacuum realizing (5)

det f / 0, (*i)*f/'.P/>~ O. (6)

Here and hereafter we use the abbreviations such as

f i -- f i (v ' v*) = 0f(q), ~)/Sq)il(~i=oi,~i:v.i ,

f / E fi](O, O*) = ~}2f(q), q))/~tOq)jii=vi,~i=o,i ,

etc. (7)

The generators X A of G(C) that leave the vacuum invariant, i.e.,

[q)i, XA]lea:o = A/A() = 0, (8)

define the unbroken symmetry subgroup H(C) of G(C), and the other generators corresponding to G(C)/H(C) change the VEV

[i, xA ] [~=o = AA() :/: O, (9)

and are called broken generators.

Theorem 1. Consider the system (I) satisfying (6) and assume that the supersymmetry remains unbroken. If the vacuum breaks the superpotential symmetry G(C) down to H(C), there must appear massless (pseudo Nambu-Goldstone) supermultiplets, in one to one correspondence to the broken generators X A . Namely, the number of the appearing massless super- multiplets is the same as the number of broken gener- ators, dim(G(C)/H(C)) = dim G(O) - dim H(C).

Proof. The equations of motion read as

- ~DB~f/oq)i + og/oq)i + o. (lO)

By multiplying Aft(q)) to this and using the symmetry (3) of superpotential g(q)), we immediately obtain from (lO)

0 = (DD~f/Oq)i)A A (q))

= [(02f/Oq)ioCk)DD~ k

+ (O3f/Oq)iO~kOCt)(D~k)(D&~l)] A/A (q)). (11)

Now recall that the VEV v i = (q)i(x, 0)) is independent ofx and 0 by the assumption of unbroken supersym- metry; i.e.,

D,v i = D&o i = O . (12)

Therefore, by the substitution of

q)i(x, O) = q)~(x, O) + o i , (13)

into (11), eqs. (11) reduce to the following simple form in the linear approximation in q)~:

.fj A A (V) DD~'] = 0. (14)

These imply that the fields

CA)(x, 0) - (A~(o))*~/q)~(x, 0) -----(~xA(o), ') , (IS)

are massless (chiral) supermultiplets at the tree level unless they vanish identically. Since the "metric" f / i s non-singular, eqs. (15) clearly show that for each one of broken generators X A (for which AA(v) 0) there appear a massless supermultiplet q)(A), say (pseudo-) Nambu-Goldstone supermultiplet. So the number of independent Nambu-Goldstone supermultiplets q)(A) is identical with the number of broken generators, dim G(C) - dim H(C). The eqs. (14) say the massless- ness of q)(A) only at the tree level approximation. But, owing to the assumption of unbroken supersymmetry, their masslessness is guaranteed by the nonrenormaliza- tion theorem [6] at any order of perturbation theory. q.e.d.

Remarks. (i) In this theorem dim G(C) (dimH(C)) means dimcG(12)(dimcH(C)) , i.e., the dimension of G(C)(H(C)) regarded as a complex Lie group. This should be so because the chiral Nambu-Goldstone superfields q)(A) are complex and hence the number of independent modes of(15) has to be counted by the number of linear independent vectors among AA(o)

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regarded as complex vectors. (ii) The Nambu-Goldstone superfields ~b (A) contain 2 [dim G(C) - dim H(C)] real bosons as their first components. Clearly these real bosons are Nambu-Goldstone bosons directly as- sociated with the broken generators X A and iX A (i: imaginary unit) of a real Lie group (G(C)) R which is equal to G(C) as a set but regarded as a group with real coordinates. The more precise correspondence of real bosons Re q~(A)(x, 0 = 0) and Im ~)(A)(x, 0 = 0) and broken generators X A and iX A would be easily seen if we examine the field-independent terms in the trans- formation laws of q~(A)

6 (e)(9 (A) = (AA (o), eBAB((~'+ o)) (16)

and separate their real and imaginary parts. Theorem 1, in its appearance, might be regarded as

a supersymmetric extension of Weinberg's pseudo- Goldstone theorem [7] in which the symmetry of the potential term V() is relevant. One should, however, notice that the superpotential symmetry neither im- plies nor is implied by the potential symmetry in general. Indeed the relation between the potential V(~o) and superpotential g(~b) is not so simple:

_ Og(~0)* [~g(~0)'~, g( tp) - ( -~ i ) [f-l(~0, tp*)] I. (17)

where ~0(x) = ~(x, 0 = 0) and (f-1)~ is the inverse ma- trix of b2f(~0, ~0*)/O(~i ~*/" Furthe/the assumption that the supersymmetry is unbroken is essential in the present theorem.

This theorem 1 is in essence identical with the one which was given by Lerche in ref. [5]. A similar state- ment was also made by Lee and Sharatchandra [4]. But here we have given it in a rather general form and made much clearer the proof and the conditions under which it holds.

Next we proceed to the general cases in which the gauge-vector multiplets Va(x, O) are also present. The lagrangian now takes the following form in general ,2.

g = fd2Od2Oy(, ~eg v)

+ ( f d20~hab()e~ aw~lV~ + h.c. )

+ ( f d2Og(~) + h.c.), (18)

where (V) /= Va(Ta)/ is the gauge vector multiplets of a gauge group Gloc. with Ta's being representation matrices of the generators X a of Gloc., and Wa

a - W~T a = -(4g~-lDfi(e-g~Dc~eg'V ) is the field- strength spinor-superfield ,3. The Gloc.-gauge in- variance of the lagrangian requires that the functions f ( , ~) (real) and g(40 satisfy the conditions

(Of/O~i)(Ta)/(~i - ~i(ra)/(Of/~i) = 0 , (19)

(Og/Oc~i)(Ta)Jd~ / = 0 , (20)

and that hab( ) transforms as a symmetric product of two adjoint representations of Gloc..

The symmetry group G(C) of the superpotential g(~) is defined by (2) and (3) also here and necessarily contains Gloc. as a (linearly realized) subgroup as is seen from (20): G(C) D Gloc.. [Here one should notice that Gloc. is in fact a complex Liegroup since the gauge transformation parameters in supersymmetric theory are chiral superfields which are complex.] We denote the generators of G(C) generally by X A with capital superfix A and the generators of Gloc. in particular by X a with lower case a. We use a symbolic notation like X A E G(X A ~ G) to indicate that the generator X A is (is not) contained in the Lie algebra of G.

Theorem 2. Consider the general system (18) with non-singular and positive matricesf/(v, v*) and Re hab(O), and assume that the supersymmetry remains unbroken. I f the superpotential symmetry G(C) is spontaneously broken down to H(C), a massless Nambu-Goldstone supermultiplet appears correspond- ing to each one of broken generators X A c G(C) such that

4:2 We have assumed in (18) that the field-strength superfields Wg appear only quadratically. But the inclusion of their higher order terms give no changes to our discussion below.

4:3 The gauge group Gloc. is not necessarily simple. Then the gauge coupling constant g"is a diagonal matrix being con- stant only within each simple (or abelian) subgroup of Gloc. But we write it for notational simplicity as if it were a constant over the whole group Gloc.

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[ePi, XAl[= o =AA(o)~0 andX A ~Gloe . . (21) So the number of the independent Nambu-Goldstone multiplets is given by

dim G(C) - dim H(C) - [dim Gloc. - dim(Glo c. ~ n(e) ) ] .

(22) Here dim Glue. - dim(Glue, f) H(C)) is the number of "would-be" Nambu-Goldstone multiplets which are actually eaten by the gauge-vector multiplets to form massive vector supermultiplets.

Proof. This theorem is just the naturally expected one is the previous theorem and the Higgs phenomenon are known. So the proof goes quite similarly to the usual proof of the Higgs theorem [8].

First, in order to obtain the linearized equations of motion, let us expand the lagrangian (18) up to qua- dratic terms in dp~ = ~i - vi and va:

= fd20d20 [(q~', ') +g(Vu, ;') +g(ck', Vv)

+ ~2(Vo, Vo) + }(Re hab ) VaDD2DV b ] (23)

+ (superpotential term) + total derivative terms.

Here we have used the property (19) o f f and the nota- tions hab = hab (o) and

((p, o~) - ~if/tI,}, (Vo)i = Va(Ta)/o i " (24)

The linearized equations of motion following from (23) are:

- ~DDf/(~b' + ~,Vo)/

+ (3g((J + o)/OdPi)linear terms in ' = 0 , (25)

(Re hab)DD2DVb +~,(Tao, ~')

+g,(ck', Tao) +g2(Tao, rbo) V b = 0 . (26)

Just as before, eq. (25) and the superpotential sym- metry (3) lead to

those modes (A) mix with the vector modes V a. As is seen further from (26), it is only the modes

ck (a) = (Tar, ok') = ( AA=a(v), ok') (28)

among 4~(A)'s that actually mix with V a. These fields represent the (dim G loc . - dim(H fh Gloc.)) indepen- dent modes corresponding to the broken generators X a contained in the gauge group Gloc.. They can be gauged away. We can simply impose ,4

(Tar, q~') = 0 , (29)

as gauge fixing conditions. Indeed, since (29) has the same form as the "unitary gauge" conditions in the usual (non-supersymmetric) Higgs case [8], the usual reasoning is already sufficient to show that (29) can be imposed. [Or equivalently, one could show that those chiral modes q~(a) of (28) are absorbed into the vector multiplets V a by the gauge transformation with chiral parameters dependent on ~(a)'s.] In the gauge (29), eq. (26) reduces to ,s

~-DD2DV a + 2(M2) ab V b = 0, (30a)

(M2) ab = ~g2(Tao, Teo)(Re h)e 1 . (30b)

These eqs. (30) show that actually (dim Gloc. - dim(H A Gloc.)) vector multiplets V a corresponding to broken generators X a E Gloc. with Tau 4~ 0 acquire non-zero masses in coincidence with the "absorption" of the chiral modes q~(a) in (28), while the other V a corresponding to the unbroken generators X a E (Gloc.

H) remain massless. On the other hand, the other chiral modes q~(A)

of (15) corresponding to broken generators X A not contained in Gloc. still represent physical Nambu- Goldstone supermultiplets as before. In fact, since D2D = 0, it follows from eqs. (30) in this gauge that

(Tao, Tbu)DDV b = 0,

or equivalently, owing to the positivity (6) o f f / ,

+4 This "supersymmetric unitary gauge" in fact fixes the gauge only in the sector of broken generators. One can still impose, for instance, DD V a = 0 for the indices a of unbroken generators.

Remarks. (i) Theorem 2 as well as theorem 1 are valid even for such "trivial cases" that the superpoten- tial g() is identically zero. When g(O) = O, any large group can be said to be the symmetry group G(C) of the superpotential term. But, as far as linear symmetry transformations are concerned, it is sufficient to take G(C) = GL(N; C) for the system containingN chiral supermultiplets i (i = 1,2 . . . . . N ) since GL(N; C) is the largest group to be realized linearly and faithful- ly on i.

(i) In this theorem 2, we have assumed the existence of non-singular kinetic term for the gauge vector multi- plets V a. But the number counting of the Nambu- Goldstone modes in theorem 2 itself remains true with- out such an assumption (e.g., even for the cases with no kinetic term of va).

To conclude this letter, we discuss two simple ex- amples which are probably instructive. We consider hereafter only symmetries of linear transformations.

As an example of theorem 1, we consider the lag- rangian

= f d2Od20(VI'i'I'i + ~i(Oli + qg~gP2i + XX)

+ ( f d20(g~iliX + m~idP2i + #(X-V)2)+h.c. ) (33)

where the index i runs over 1-n. This is the model dis- cussed first by Buchmialler et al. [9] and later by other authors [4,5] for the case ofn = 2. The symmetry of the lagrangian is G O = SU(n) X U(1), under which the superfields 4 , q~l,2 and X transforms as (n*, -1) , (n, +1) and (1, 0). But the symmetry of the superpoten- tial is G(C) = GL(n;C). Under this GL(n; C), xI ti and (q~l,2)i are contravariant and covariant n-vectors, respec- tively, and X is scalar.

The vacuum candidates at the tree level corre- spond to the minima of the potential V() (17). As is easily seen, a supersymmetric minimum is realized at

(~oi} = 0 , (1i) = mC/, (~b2i> = -goC i , (X) = o, (34a)

where C/is an arbitrary constant n-vector but can be generally brought into the form

Goldstone bosons corresponding to the breaking of the lagrangian symmetry SU(n) down to SU(n - 1). On the other hand, for the diagonal part i = n, iX n is a U(1) generator of G O but X n (or equivalently the trace part 2n= 1 X[) is a generator of scale transformation not belonging to G 0. So Im (n,n)(o = 0) is a true Nambu- Goldstone boson but Re (a(n,n)(o = 0) is a pseudo Nambu-Goldstone boson associated with a breakdown of scale transformation symmetry of superpotential 14,5].

As an example of theorem 2, we discuss the follow- ing model with zero superpotential g(~b) = 0:

2= fdZOd2b[~i (egV) /~ - g-'V/l , (39)

where V(n X n matrix) is a gauge vector multiplet of gauge group Gloc. = U(n) ,6 and q~a(/= 1 -n, a = l -m) belongs to a representation n of Gloc. and m* of a global U(m). The lagrangian symmetry is Gloc. X global U(m). We assume n < m. This model (39) was construct- ed by Aoyama [10] as a straightforward generalization of the supersymmetric CP n - 1 model considered by D'Adda, Di Vecchia and Ltischer [ 11 ] in two-dimen- sional space-time. As was remarked above, it is suf- ficient to take G(C) = GL(nm;C) as the superpotential symmetry since ~b a has nm components. If we define the generators Y /andZa b of GL(n; C) and GL(m; C) acting on the indices i and a of Ct .a, respectively, in a similar way to the Xj t in the preceding example, then, the (nm) 2 generators of GL(nm; C) may be given by Yj zab :

[e) a, Yj; Zab, ' ] =(8f@)~bS.b (8b'sa,) = 61'

Volume 135B, number 5,6 PHYSICS LETTERS 16 February 1984

Note added in proof. In some explicit examples, Kubo and Sakakibara [14] counted the numbers of Nambu-Goldstone supermultiplets in the same way as our theorems 1 and 2. We thank them for calling our attention to their work.